11 THE SPACES L1 AND L2 165
sequence of functions consisting of 1 and /'" (n J> 1), where lim an = -f oo , is a total
sequence in the space ^n(I), if and only if ]T a^1 = + co ("Muntz's theorem").
In -^R(X, p) or ^c(X, /*) a sequence (/) of functions is said to be orthonormal if
the sequence of classes fn of these functions is an orthonormal system in the corre-
sponding Hilbert space.
(a) Let (fn) be an infinite orthonormal sequence, uniformly bounded on X. If p,
is bounded, show that there exists a set A with measure >0 such that£ \fn(x)\2 = + oo
n
for all x e A. (Argue by contradiction, observing that the sequence (fn) cannot tend
to 0 almost everywhere.)
(b) Suppose that p, is bounded. Let (/) be a uniformly bounded infinite orthonormal
sequence. If a sequence (an) of scalars is such that the series with general term anfn(x)
converges almost everywhere, show that lim an = 0 (use EgorofTs theorem applied to
H-+00
the sequence (\anfn\2)). Show that there exists a sequence of scalars (bn) such that
lim bn = 0 and ]T I bn \ 2 \fn(x) \2 = + oo on a set of measure >0. (Choose the sequence
n-*oo «
(bn) so that ]£ | bn \ 2 = + oo , and argue by contradiction, by applying EgorofTs
n
theorem to the partial sums of the series ]T | bn \ 2 \fn \ 2.)
n
(c) Let (/) be a sequence in &2. Show that the following three conditions are
equivalent :
a) If a sequence (an) is such that the series with general term anfn(x) converges
on a set of measure >0, then lim an = 0.
n-+oo
ft) If a sequence (an) is such that the series with general term anfn(x) converges
absolutely on a set of measure >0, then lim an 0.
n-*oo
y) For every measurable set A of measure >0, we have
liminf \fn\ dp>0.
w-foo JA
(To prove that y) implies a) and ft), use EgorofF s theorem. To prove the converse,
argue by contradiction , by considering a subsequence (fnk) such that | fnk \ dp, > 1 Ik 3,
and taking ank k and an 0 if n 9*= nk .)
(d) Suppose that ^ is bounded and (/) is a sequence in 3? 2. Prove that the following
three conditions are equivalent :
a) For each sequence (an) such that the series with general term anfn(x) con-
verges almost everywhere, we have lim an 0.
n~+ oo
/?) For each sequence (an) such that the series with general term anfn(x)
converges absolutely almost everywhere, we have lim an 0.
n-*oo
y) There exists 8 > 0 such that, for every measurable set A whose complement
has measure <|S, we have lim inf |/J dju. > 0. (Same method as (c).) Deduce that,
H_KJO JA
if the sequence (/) is orthonormal and uniformly bounded, it satisfies these three
conditions, and therefore the series with general term aHfn(x) converges absolutely
almost everywhere if and only if 2] |an| < + oo (use (b)).ontinuous affine linear functions on E is a subspace of the separable